I've wanted something more like Emacs for the web -- a bunch of fast primitives and hooks everywhere, with a good-enough language to allow me to modify everything about the browser as I see fit, without having to drop down into the C++ hex mines.
Seriously, though, I want the software I run to obey me, not someone else: not Microsoft, not Apple, not some government on the other side of the world — and not Mozilla. I want to be able to modify its functionality in a straightforward manner, in a reasonably-sane garbage-collected language (so, not C, not C++, not Java and not JavaScript).
I would like to be able to add pull data from sites I visit and plumb it through the rest of my system using something like Plan 9's plumber. I don't want a browser which says, 'sorry, I won't trust you to choose the extensions you like.'